0
votes
1answer
29 views

Canonical Separation of variables

Do the functions of the form $\psi(x)\phi(y)$ span $L^2(\mathbf{R}^6)$? Insert proper grammar here.
8
votes
1answer
200 views

General formula needed for this product rule expression (differential operator)

Let $D_i^t$, $D_i^0$ for $i=1,\dots,n$ be differential operators. (For example $D_1^t = D_x^t$, $D_2^t = D_y^t,\dots$, where $x$, $y$ are the coordinates). Suppose I am given the identity $${D}_a^t ...
0
votes
1answer
48 views

Bounding the integral of a $C^1$ function

Supopse that $\Psi \in C^1(\mathbb{R},\mathbb{R})$, $\Psi(x) \geq - C_0 \, \forall x$ and $|\Psi'(x)| \leq C_1|x|^q+C_2 \, \forall x$. Let $u \in H^1(\Omega)$. Under these assumptions how can I bound ...
2
votes
1answer
53 views

Is this function in $L^2(\mathbb{R}^6)$?

I have to prove that the following function in $L^2(\mathbb{R}^6)$ $$F(x,y)=\frac{f(x)}{x^2+y^2+\frac{2}{m+1}x\cdot y+\lambda}$$ with $f\in H^{\frac{1}{2}}(\mathbb{R}^3)$, $x,y\in\mathbb{R}^3$ and ...
1
vote
1answer
55 views

How to establish the estimate?

I consider the inequality as follows: let $a>0,b>0$, and satisfy $$2a^2-b^2\leq C(1+a).\tag{1}$$ If we assume that $b\leq a$, then there exist a constant $C_1>0$ such that $$a\leq ...
2
votes
1answer
43 views

How to esimate $\inf\int|\nabla g|^p\,dx$

It is rather easy question but I'm already struggling with this problem for a long time. I'm trying to estimate the value $$\inf\int|\nabla g|^p\,dx$$ where $\mathbf{inf}$ is taken over all ...
0
votes
0answers
92 views

Integrating $\ln(1+|\ln|x||)$ in $B_1(0)$

I am trying to integrate $$\int_{B_1(0)} \ln(1+|\ln|x||).$$ $B_1(0)\subset \mathbb R^n$ Basically what I am trying to see is whether $\ln(1+|\ln|x||)$ belongs to $L^\infty (B_1(0))$ and ...
4
votes
1answer
163 views

Not so obvious calculus question

You have $f \in C^\infty([0,1])$ with $f > 0$. Then $\sqrt{f}$ is easily seen to be differentiable . Prove that there exists a constant $C$ independent of $f$ such that: ...